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Bickart Points

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BickartPoints

The Bickart points are the foci F_1 and F_2 of the Steiner circumellipse. They have trilinear coordinates alpha_1:beta_1:gamma_1 and alpha_2:beta_2:gamma_2, where

alpha_i=(sqrt(2)Delta)/a+/-sqrt(-a^2+Z+bccos(B-C))
(1)
beta_i=(sqrt(2)Delta)/b+/-sqrt(-b^2+Z+accos(A-C))
(2)
gamma_i=(sqrt(2)Delta)/c+/-sqrt(-c^2+Z+abcos(A-B))
(3)

where

Z=sqrt(a^4+b^4+c^4-a^2b^2-b^2c^2-c^2a^2).
(4)

and for some appropriate choices of signs. These can be rewritten in fully closed form as

alpha_+/-=(Q-k(b^2-c^2)(a^4-b^2c^2-a^2Z))/a
(5)
beta_+/-=(Q-k(c^2-a^2)(b^4-c^2a^2-b^2Z))/b
(6)
gamma_+/-=(Q-k(a^2-b^2)(c^4-a^2b^2-c^2Z))/c
(7)

where k=2 and

Q=sqrt(2(a^2b^2c^2Z^3)-2[b^4c^4(2S^2-a^2S_omega)+c^4a^4(2S^2-b^2S_omega)+a^4b^4(2S^2-c^2S_omega)])
(8)

(P. Moses, pers. comm., Mar. 31, 2006), where S_omega is Conway triangle notation.

The foci of the Steiner inellipse are given by the same equation, but instead taking k=1.

See also

Circumellipse, Focus, Ellipse, Steiner Circumellipse

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References

Castellsaguer, Q. "Bickart Points." http://www.xtec.es/~qcastell/ttw/ttweng/definicions/d_Bickart_p.html.Yiu, P. Introduction to the Geometry of the Triangle. Florida Atlantic University lecture notes, Version 2.0402, p. 129, April 2002. https://web.archive.org/web/20060903181437/http://www.math.fau.edu/Yiu/GeometryNotes020402.pdf.

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Bickart Points

Cite this as:

Weisstein, Eric W. "Bickart Points." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BickartPoints.html

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